user's guide dpin
TRANSCRIPT
DPIN 3.0
A PROGRAM FOR DECOMPOSING PRODUCTIVITY
INDEX NUMBERS
by
C.J. O’Donnell
The University of Queensland
Centre for Efficiency and Productivity Analysis
Brisbane 4072, Australia
email: [email protected]
20 September 2011
1. Introduction ..................................................................................................................................... 2
2. Total Factor Productivity Indexes .................................................................................................. 2
3. Measures of Efficiency .................................................................................................................... 4
4. The Components of TFP Change ................................................................................................... 7
5. Estimation Using DEA .................................................................................................................... 8
Productivity Indexes ........................................................................................................................... 8
Technical and Scale Efficiency ........................................................................................................... 9
Mix Efficiency .................................................................................................................................... 9
Other Efficiency and Productivity Measures .................................................................................... 10
Shadow Prices ................................................................................................................................... 11
6. Installing and Running the DPIN Software ................................................................................. 12
Creating the DPIN Input File ............................................................................................................ 12
Running the Executable File ............................................................................................................. 14
7. DPIN Output Files ......................................................................................................................... 17
8. Examples ........................................................................................................................................ 18
9. Runtime Errors .............................................................................................................................. 20
9. References ...................................................................................................................................... 27
2
1. INTRODUCTION
An index number is a measure of change in a variable, or group of variables, over time or space. The DPIN
computer program uses the aggregate-quantity framework developed by O'Donnell (2008) to compute and
decompose productivity index numbers. The O'Donnell (2008) methodology does not rely on the availability of
price data and does not require any assumptions concerning either the degree of competition in product markets
or the optimizing behaviour of firms. Thus, DPIN can be used to analyse the drivers of productivity change
even when prices are unavailable and/or industries are non-competitive. The program uses data envelopment
analysis (DEA) linear programs (LPs) to estimate the production technology and levels of productivity and
efficiency. The program then decomposes changes in productivity into measures of
(a) technical change (measuring movements in the production frontier);
(b) technical efficiency change (movements towards or away from the frontier);
(c) scale efficiency change (movements around the frontier surface to capture economies of scale); and
(d) mix efficiency change (movements around the frontier surface to capture economies of scope).
This Guide outlines the methodological framework and provides a guide to installing and running the program.
2. TOTAL FACTOR PRODUCTIVITY INDEXES
The productivity of a single-output single-input firm is almost always defined as the output-input ratio.
O'Donnell (2008) generalizes this idea to the multiple-output multiple-input case by formally defining the total
factor productivity (TFP) of a firm to be the ratio of an aggregate output to an aggregate input. Let
1( ,..., )it it Kitx x x and 1( ,..., )it it Jitq q q denote the input and output quantity vectors of firm i in period t. Then
the TFP of the firm is
(1)
itit
it
QTFP
X (total factor productivity)
where ( )it itQ Q q is an aggregate output, ( )it itX X x is an aggregate input, and (.)Q and (.)X are non-
negative, non-decreasing and linearly homogeneous aggregator functions. The associated index number that
measures the TFP of firm i in period t relative to the TFP of firm h in period s is
(2) ,
,,
/
/ hs itit it it
hs iths hs hs hs it
QTFP Q XTFP
TFP Q X X (TFP index)
where , /hs it it hsQ Q Q is an is an output quantity index and , /hs it it hsX X X is an input quantity index. Thus,
TFP growth can be expressed as a measure of output growth divided by a measure of input growth.
Different aggregator functions give rise to different TFP indexes. The class of non-negative, non-decreasing
and linearly homogeneous aggregator functions includes
3
(3) ( ) hsQ q p q (Laspeyres)
(4) ( ) itQ q p q (Paasche)
(5) 1/ 2( ) ( )it hsQ q p qq p (Fisher)
(6) 0( )Q q p q (Lowe)
(7) ( ) ( , , )O hsQ q D x q s (Malmquist-hs)
(8) ( ) ( , , )O itQ q D x q t (Malmquist-it)
(9) 1/ 2( ) [ ( , , ) ( , , )]O hs O itQ q D x q s D x q t (Hicks-Moorsteen)
(10) 0 0( ) ( , , )OQ q D x q t (Färe-Primont)
(11) ( ) hsX x w x (Laspeyres)
(12) ( ) itX x w x (Paasche)
(13) 1/ 2( ) ( )hs itX x w xx w (Fisher)
(14) 0( )X x w x (Lowe)
(15) ( ) ( , , )I hsX x D x q s (Malmquist-hs)
(16) ( ) ( , , )I itX x D x q t (Malmquist-it)
(17) 1/ 2( ) [ ( , , ) ( , , )]I hs I itX x D x q s D x q t (Hicks-Moorsteen) and
(18) 0 0( ) ( , , )IX x D x q t (Färe-Primont)
where 1( ,..., )it it Kitw w w and 1( ,..., )it it Jitp p p are vectors of input and output prices; 0 ,p 0 ,w 0q and 0x are
vectors of representative prices and quantities; t0 denotes a representative time period; and (.)OD and (.)ID are
Shephard (1953) output and input distance functions. The aggregator functions (3) to (18) are so-named
because when they are substituted into (1) and (2) they give rise to the following TFP indexes:
(19) ,hs it hs hs
hs iths hs hs it
p q w xTFP
p q w x
(Laspeyres)
(20) ,it it it hs
hs itit hs it it
p q w xTFP
p q w x
(Paasche)
(21)
1/ 2
,it it hs it hs hs it hs
hs itit hs hs hs hs it it it
p q p q w x w xTFP
p q p q w x w x
(Fisher)
(22) 0 0
,0 0
it hshs it
hs it
p q w xTFP
p q w x
(Lowe)
(23) ,
( , , ) ( , , )
( , , ) ( , , )O hs it I hs hs
hs itO hs hs I it hs
D x q s D x q sTFP
D x q s D x q s (Malmquist-hs)
(24) ,
( , , ) ( , , )
( , , ) ( , , )O it it I hs it
hs itO it hs I it it
D x q t D x q tTFP
D x q t D x q t (Malmquist-it)
(25)
1/ 2
,
( , , ) ( , , ) ( , , ) ( , , )
( , , ) ( , , ) ( , , ) ( , , )O hs it I hs hs O it it I hs it
hs itO hs hs I it hs O it hs I it it
D x q s D x q s D x q t D x q tTFP
D x q s D x q s D x q t D x q t
(Hicks-Moorsteen) and
(26) 0 0 0 0
,0 0 0 0
( , , ) ( , , )
( , , ) ( , , )O it I hs
hs itO hs I it
D x q t D x q tTFP
D x q t D x q t (Färe-Primont).
4
The Laspeyres, Paasche and Fisher indexes defined by (19) to (21) are well-known in the productivity literature.
O'Donnell (2010b) refers to the index (22) as a Lowe TFP index because the component output quantity and
input quantity indexes have been traced back to Lowe (1823). This Guide refers to the indexes (23) and (24) as
Malmquist-hs and Malmquist-it indexes because the component output quantity and input quantity indexes are
the firm-specific Malmquist indexes defined by Caves, Christensen and Diewert (1982, p. 1396,1400). The
index defined by (25) was first proposed by Bjurek (1996) but is commonly known as a Hicks-Moorsteen index
because it is the geometric average of two indexes that Diewert (1992, p. 240) attributed to Hicks (1961) and
Moorsteen (1961). Finally, the index defined by (26) was first proposed by O'Donnell (2011a) but is referred to
in this Guide as a Färe-Primont index because it can be written as the ratio of two indexes defined by Färe and
Primont (1995).
Lowe and Färe-Primont indexes are economically-ideal in the sense that they satisfy all economically-relevant
axioms and tests from index number theory, including an identity axiom and a transitivity test. This means they
can be used to make reliable multi-temporal (i.e., many period) and/or multi-lateral (i.e., many firm) compari-
sons of TFP and efficiency. Laspeyres, Paasche, Fisher, Malmquist-hs, Malmquist-it and Hicks-Moorsteen
indexes all fail the transitivity test and can generally only be used to make a single binary comparison (i.e., to
compare two observations only). For more details on the importance of index number axioms and tests, see
O'Donnell (2011b).
3. MEASURES OF EFFICIENCY
Most, if not all, economic measures of efficiency can be defined as ratios of measures of TFP. Examples
include (O'Donnell (2008))
(27) *1it
itt
TFPTFPE
TFP (TFP efficiency)
(28) /
( , , ) 1/
it it itit O it it
it it it
Q X QOTE D x q t
Q X Q (output-oriented technical efficiency)
(29) /
1/
it it
it
it it
Q XOSE
Q X (output-oriented scale efficiency)
(30) /
1ˆ ˆ/
it it itit
it it it
Q X QOME
Q X Q (output-oriented mix efficiency)
(31) *
ˆ /1it it
itt
Q XROSE
TFP (residual output-oriented scale efficiency)
(32) 1/
( , , ) 1/
it it itit I it it
it it it
Q X XITE D x q t
Q X X (input-oriented technical efficiency)
(33) /
1/
it it
it
it it
Q XISE
Q X (input-oriented scale efficiency)
5
(34)
ˆ/1
ˆ/ it it it
ititit it
Q X XIME
XQ X (input-oriented mix efficiency)
(35) *
ˆ/1it it
itt
Q XRISE
TFP (residual input-oriented scale efficiency) and
(36) *
/1it it
itt
Q XRME
TFP
(residual mix efficiency)
where *tTFP denotes the maximum TFP that is possible using the technology available in period t;
1( , , )it it O it itQ Q D x q t is the maximum aggregate output possible when using itx to produce a scalar multiple of
;itq 1( , , )it it I it itX X D x q t is the minimum aggregate input possible when using a scalar multiple of itx to
produce ;itq ˆ
itQ is the maximum aggregate output possible when using itx to produce any output vector; ˆitX is
the minimum aggregate input possible when using any input vector to produce ;itq and itQ and itX are the
aggregate output and input obtained when TFP is maximized subject to the constraint that the output and input
vectors are scalar multiples of itq and itx respectively.
The technical efficiency measures given by (28) and (32) are usually attributed to Farrell (1957). The scale
efficiency measures given by (29) and (33) are the conventional measures defined by, for example, Balk (1998,
p. 20, 23). The remaining measures of efficiency were first defined by O'Donnell (2008) – TFP efficiency is a
measure of overall productive performance, while measures of residual scale and mix efficiency are measures of
productive performance associated with economies of scale and scope. Other important measures of efficiency
include (O'Donnell (2010b))
(37) 1it it it it itOSME OME ROSE OSE RME (output-oriented scale-mix efficiency) and
(38) 1it it it it itISME IME RISE ISE RME (input-oriented scale-mix efficiency).
To illustrate the relationship between measures of productivity and efficiency, several of the measures defined
by (27) to (38) are depicted in Figures 1 and 2. In these figures, the curve passing through point D is what
O'Donnell (2008) refers to as a mix-restricted frontier – it is the boundary of the set of all technically-feasible
aggregate input-output combinations that have the same input and output mix as the firm operating at point A.
The curve passing through point E is an unrestricted production frontier – it is the boundary of the production
possibilities set that is available to firms when all mix restrictions are relaxed. O'Donnell (2008) shows how
measures of TFP and efficiency can be expressed in terms of slopes of rays in aggregate quantity space. For
example, the TFP of the firm operating at point A in Figure 1 is / slope 0A,it it itTFP Q X the measure of
TFP efficiency defined by (27) is */ slope 0A / slope 0E,it it tTFP TFP TFP and the measure of residual
output-oriented scale efficiency defined by (31) is *ˆ( / ) / slope 0V / slope 0E.it it it tROSE Q X TFP
For more
details see O'Donnell (2008).
6
Figure 1. Output-Oriented Measures of Efficiency for a Multiple-Input Multiple-Output Firm
Figure 2. Input-Oriented Measures of Efficiency for a Multiple-Input Multiple-Output Firm
A
Aggregate Output
Aggregate Input
0
itQ
itXitX
B
U
ˆitX *
tX
*tQ E
itX
D itQ
scale inefficiency
mix inefficiency
scale-mix inefficiency
technical inefficiency
residual mix inefficiency
residual scale inefficiency
A
Aggregate Output
Aggregate Input
0
itQ
itX
itQ C
*tX
*tQ E
itX
D itQ
residual mix inefficiency
scale-mix inefficiency
technical inefficiency
ˆitQ mix
inefficiency
residual scale inefficiency scale
inefficiency
V
7
Associated with the aggregate output and input quantities itQ and itX are (implicit) aggregate output and input
prices /it it it itP p q Q and / .it it it itW w x X If prices are available then DPIN can also be used to compute the
following measures of efficiency:
(39) 1( , , )
it itit
it it
P QRE
r x p t (revenue efficiency)
(40) 1( , , )
it itit
it it
P QRAE
r x p t (revenue-allocative efficiency)
(41) ( , , )
1it itit
it it
c w q tCE
W X (cost efficiency) and
(42) ( , , )
1it itit
it it
c w q tCAE
W X (cost-allocative efficiency)
where ( , , )it itr x p t is the maximum revenue that can be obtained in period t when the input vector is itx and the
output price vector is ,itp and ( , , )it itc w q t is the minimum cost of producing itq in period t when the input price
vector is .itw
4. THE COMPONENTS OF TFP CHANGE
O'Donnell (2008) refers to TFP indexes that can be expressed in terms of aggregate quantities as in equation (2)
as being multiplicatively-complete. All such TFP indexes can be decomposed into a measure of technical
change and various measures of efficiency change. The simplest way to see this is to rewrite equation (27) as* .it t itTFP TFP TFPE
A similar equation holds for firm h in period s: * .hs s hsTFP TFP TFPE It follows that
the TFP index (2) can be decomposed as
(43)
*
, *.t it
hs ithss
TFP TFPETFP
TFPETFP
The first term in parentheses on the right-hand side of (43) measures the change in the maximum TFP over time
– this is a natural measure of technical change. The second term is a measure of overall efficiency change.
Equations (28) to (42) can be used to effect an even finer decomposition of TFP change than the simple decom-
position given by equation (43). For example, three finer output-oriented decompositions are
(44)
*
, *t it it
hs iths hss
TFP OTE OSMETFP
OTE OSMETFP
(45)
*
, *t it it it
hs iths hs hss
TFP OTE OSE RMETFP
OTE OSE RMETFP
and
(46)
*
, *.t it it it
hs iths hs hss
TFP OTE OME ROSETFP
OTE OME ROSETFP
8
5. ESTIMATION USING DEA
If prices are available then computing Laspeyres, Paasche, Fisher and Lowe indexes is straightforward using
equations (19) to (22). However, decomposing these indexes into measures of technical change and efficiency
change involves estimating the production technology. Whether or not prices are available, estimating (and
decomposing) Malmquist-hs, Malmquist-it, Hicks-Moorsteen and Färe-Primont indexes also involves estimating
the technology. DPIN estimates the production technology (and associated measures of productivity and
efficiency) using DEA LPs. DEA is underpinned by the assumption that the (local) output and input distance
functions representing the technology available in period t take the form (e.g., O'Donnell (2011b))
(47) ( , , ) ( ) /( )O it it it itD x q t q x and
(48) ( , , ) ( ) /( ).I it it it itD x q t x q
The standard output-oriented DEA problem involves selecting values of the unknown parameters in (47) in
order to minimize 1 1( , , ) .it O it itOTE D x q t The input-oriented problem involves selecting values of the
unknown parameters in (48) in order to maximise 1( , , ) .it I it itITE D x q t The resulting linear programs are
(e.g., O'Donnell (2011b))
(49) 1 1
, ,( , , ) min : ; 1; 0; 0O it it it it itD x q t OTE x X Q q
and
(50) 1
, ,( , , ) max : ; 1; 0; 0I it it it it itD x q t ITE q Q X x
where Q is a tJ M matrix of observed outputs, X is a tK M matrix of observed inputs, ι is an 1tM unit
vector, and tM denotes the number of observations used to estimate the frontier in period t. DPIN uses
variants of these two LPs to compute productivity indexes and measures of efficiency (change).
Productivity Indexes
DPIN estimates Malmquist-hs, Malmquist-it, Hicks-Moorsteen and Färe-Primont aggregates by first solving the
following variants of LPs (49) and (50) (O'Donnell (2011b)):
(51) 1
, ,( , , ) min : ; 1; 0; 0O hs it hs itD x q s x X Q q
(Malmquist-hs)
(52) 1
, ,( , , ) max : ; 1; 0; 0I it hs hs itD x q s q Q X x
(Malmquist-hs)
(53)
1
, ,( , , ) min : ; 1; 0; 0O it hs it hsD x q t x X Q q
(Malmquist-it)
(54) 1
, ,( , , ) max : ; 1; 0; 0I hs it it hsD x q t q Q X x
(Malmquist-it)
(55) 10 0 0 0 0
, ,( , , ) min : ; 1; 0; 0OD x q t x X Q q
(Färe-Primont)
(56)
10 0 0 0 0
, ,( , , ) max : ; 1; 0; 0ID x q t q Q X x
(Färe-Primont)
9
Aggregate outputs and inputs are then estimated as (O'Donnell (2011b))
(57) ( ) /( )it it hs hs hs hsQ q x (Malmquist-hs)
(58) ( ) /( )it it hs hs hs hsX x q (Malmquist-hs)
(59)
( ) /( )hs hs it it it itQ q x
(Malmquist-it)
(60) ( ) /( )hs hs it it it itX x q (Malmquist-it)
(61) 0 0 0 0( ) /( )it itQ q x
(Färe-Primont) and
(62)
0 0 0 0( ) /( )it itX x q
(Färe-Primont)
where ,hs ,hs ,hs ,hs hs and hs solve (51) and (52), ,it ,it ,it ,it it and it solve (53) and (54),
and 0 , 0 , 0 , 0 , 0 and 0 solve (55) and (56). DPIN uses sample mean vectors as representative
output and input vectors in LPs (55) and (56). The representative technology in these two LPs is the technology
obtained under the assumption of no technical change (i.e., tM is the sample size). Each of the LPs (51) to (56)
allows the technology to exhibit variable returns to scale (VRS). If the technology is assumed to exhibit
constant returns to scale (CRS) then DPIN sets 0.
Technical and Scale Efficiency
DPIN obtains measures of technical efficiency by solving the following dual LPs:
(63)
1
,/ ( , , ) min : ; ; 1; 0it it it O it it it itOTE Q Q D x q t q Q X x
and
(64)
1
,/ ( , , ) min : ; ; ; 0it it it I it it it itITE X X D x q t Q q x X
where is an 1tM vector. To estimate measures of technical efficiency under a CRS assumption, DPIN
removes the constraint 1 and solves
(65)
1
,( , , ) min : ; ; 0CRS
it O it it it itOTE H x q t q Q X x
and
(66)
,
( , , ) min : ; ; 0 .CRSit I it it it itITE H x q t Q q x X
Measures of scale efficiency are then computed as
(67)
/CRSit it itOSE OTE OTE and
(68)
/ .CRSit it itISE ITE ITE
Mix Efficiency
Measures of mix efficiency are defined by equations (30) and (34). Estimates of ˆ ,itQ ˆ ,itX ˆhsQ and ˆ
hsX are
obtained by solving the following LPs:
10
(69)
,
ˆ max ( ) : ; ; 1; 0it itq
Q Q q q Q X x
and
(70)
,
ˆ min ( ) : ; ; 1; 0 .it itxX X x Q q x X
For any aggregator function, LP (69) gives the maximum aggregate output that can be produced using itx (i.e.,
the maximum aggregate output that firm i in period t could produce using its input vector), while LP (70) gives
the minimum aggregate input that can produce itq (i.e., the minimum aggregate input that could be used by firm
i in period t to produce its output vector). Paasche, Laspeyres and Lowe estimates of ˆ ,itQ ˆ ,itX ˆhsQ and ˆ
hsX are
obtained by replacing ( )Q q and ( )X x in (69) and (70) with the aggregator functions (3), (4), (6), (11), (12) and
(14). Fisher estimates are obtained by taking the geometric average of the Paasche and Laspeyres estimates.
Malmquist-hs, Malmquist-it and Färe-Primont estimates are obtained by replacing ( )Q q and ( )X x with
functions (57) to (62). Hicks-Moorsteen estimates are obtained by taking the geometric average of the Malm-
quist-hs and Malmquist-it estimates.
Other Efficiency and Productivity Measures
DPIN computes the maximum TFP in period t as
(71)
* max / .t it iti
TFP Q X
Other efficiency and productivity measures are computed residually:
(72) /it it itTFP Q X
(73) */it it tTFPE TFP TFP
(74) *ˆ( / ) /it it it tROSE Q X TFP
(75) *ˆ( / ) /it it it tRISE Q X TFP
(76) it it itOSME OME ROSE
(77) it it itISME IME RISE and
(78) .itit
it it
TFPERME
OTE OSE
Again, results for Fisher and Hicks-Moorsteen indexes are computed as geometric averages of the Laspeyres,
Paasche, Malmquist-hs and Malmquist-it indexes as appropriate. If the Paasche aggregator function is used to
measure efficiency and TFP then output- and input-oriented measures of mix efficiency for firm i in period t
will be measures of revenue-allocative efficiency ( )itRAE and cost-allocative efficiency ( )itCAE respectively
(O'Donnell (2010b)). EXCEL commands can then be used to compute
(79) it it itRE OTE RAE (revenue efficiency) and
(80) .it it itCE ITE CAE (cost efficiency)
11
If prices are available then DPIN also computes
(81) /it it itP REV Q (aggregate output price)
(82) /it it itW COST X (aggregate input price)
(83) /it it itTT P W (terms of trade) and
(84) /it it itPROF REV COST (profitability)
where it it itREV p q denotes the total revenue of firm i in period t and it it itCOST w x denotes total cost.
Finally, if prices are available then DPIN decomposes profitability change into the product of an index measur-
ing the change in the terms-of-trade (i.e., ratio of output prices to input prices) and the change in TFP (i.e., ratio
of output quantity to input quantity) (e.g., O'Donnell (2010a)):
(85) , , ,
, , ,, , ,
hs it hs it hs itiths it hs it hs it
hs hs it hs it hs it
REV P QPROFPROF TT TFP
PROF COST W X
(profitability index)
where , / ,hs it it hsREV REV REV , / ,hs it it hsCOST COST COST , / ,hs it it hsP P P , /hs it it hsW W W and
, , ,/hs it hs it hs itTT P W are revenue, cost, output price, input price and terms-of-trade indexes respectively.
Shadow Prices
The derivatives of output and input distance functions with respect to outputs and inputs can be interpreted as
revenue- and cost-deflated output and input shadow prices (e.g., Grosskopf, Margaritis and Valdmanis (1995)).
For example, the first-derivatives of the (local) output and input distance functions (47) and (48) are
(86) * ( , , ) / /( )it O it it it itp D x q t q x and
(87) * ( , , ) / /( ).it I it it it itw D x q t x q
DPIN evaluates these derivatives (shadow prices) at the values of , , , , and that solve LPs (49)
and (50). By way of further example, the derivatives of 0 0 0( , , )OD x q t and 0 0 0( , , )ID x q t are
(88) *0 0 0 0 0 0( , , ) / /( )Op D x q t q x and
(89) *0 0 0 0 0 0( , , ) / /( ).Iw D x q t x q
DPIN evaluates these shadow prices at the values of , , , , and that solve LPs (55) and (56).
Observe from equations (6), (14), (61) and (62) that Färe-Primont indexes are identical to Lowe indexes
whenever *0 0p p and *
0 0 .w w Similar relationships exist between Malmquist-hs and Laspeyres indexes, and
between Malmquist-it and Paasche indexes (O'Donnell (2011b)).
12
6. INSTALLING AND RUNNING THE DPIN SOFTWARE
DPIN is written in C++ and is designed to run on a Windows XP or Vista platform. The program and associated
files can be downloaded from DPIN website at http://www.uq.edu.au/economics/cepa/dpin.htm. DPIN is
available two editions: the Standard edition will compute and decompose Malmquist-hs, Malmquist-it, Hicks-
Moorsteen and Färe-Primont TFP indexes and is available free-of-charge; the Professional edition will also
compute and decompose Paasche, Laspeyres, Fisher and Lowe TFP indexes and is available on payment of an
annual license fee. Both editions are hard-wired to analyze up to 5000 observations. Further details concerning
the functionality of the two editions and the pricing of the Professional edition are available on the DPIN
website. Irrespective of the edition, installing DPIN involves downloading a .zip file from the DPIN website
and extracting the contents of the file into any directory. Installing the Professional edition also involves
purchasing a license key by following the instructions on the DPIN website. Running DPIN then involves
creating an input file and running the executable file.
Creating the DPIN Input File
DPIN input files can be created within Microsoft EXCEL and must be saved in a .csv (comma-delimited)
format. The input file contains both commands and data. To illustrate, Figure 3 is a screenshot of the input file
Eg1_input.csv required for the example discussed in O'Donnell (2011b). The rows before the end command are
“command rows”; the row immediately after the end command is a “header row”; and the remaining rows are
“data rows”. DPIN will read the text and data in columns A and B of every row until it encounters an end
command. It will then skip the end command and the header row and start reading the data. The DPIN program
is case-sensitive and will only recognize certain text strings in the command rows. The main commands are
Firms to specify the number of firms
Periods to specify the number of periods
Outputs to specify the number of outputs
Inputs to specify the number of inputs
Prices included if and only if the input file contains prices as well as quantities
The command rows (i.e., the rows before the end command) can be listed in any order (e.g., the number of
outputs can be specified first, second, or last). The program will not read the output and input variable names in
the header row (i.e., the row immediately after the end command). The output and input data must be stored in
the data rows in a particular format:
the observation identifier must be stored in column A; the firm and period identifiers must be stored in
columns B and C; all identifiers must be numeric (e.g., 1, 2, ..., not Jan, Feb ...);
the first output quantity variable must be stored in column D; all the output quantity variables must be
stored first, followed by the input quantity variables (e.g., q1, q2, q3, x1, x2);
13
Figure 3. Example Input File for Computing and Decomposing Lowe TFP Indexes
if prices are available then the output and input quantity variables must be stored first, followed by the
output price variables, followed by the input price variables (e.g., q1, q2, q3, x1, x2, p1, p2, p3, w1, w2);
the data must be a balanced panel; if the panel is unbalanced then it can be artificially balanced by replac-
ing any missing observations with any other observations from the same time period (this will not affect
measures of OTE, ITE, OSE or ISE, but Malmquist-hs, Malmquist-it, Hicks-Moorsteen and Färe-Primont
estimates of mix efficiency and TFP may be sensitive to the choice of replacement observations);
the data must be sorted first by period and then by firm (i.e., all observations from period 1 must be
listed first, followed by all observations from period 2, and so on);
The default TFP index in the Standard edition of DPIN is the Färe-Primont index. The default index in the
Professional edition is the Lowe index when prices are included in the input file and the Färe-Primont index
otherwise. The default TFP index can be changed using one of the following commands:
Laspeyres for Laspeyres indexes (available in Professional edition only)
Paasche for Paasche indexes (available in Professional edition only)
Fisher for Fisher indexes (available in Professional edition only)
Lowe for Lowe indexes (available in Professional edition only)
Malmquist-hs for Malmquist-hs indexes
Malmquist-it for Malmquist-it indexes
HicksMoorsteen for Hicks-Moorsteen indexes
FarePrimont for Färe-Primont indexes
14
The default DPIN settings are to estimate the technology allowing for technical regress, technical progress and
variable returns to scale. These and other default settings can be changed using the following commands:
Base to specify the reference observation when computing (transitive) Lowe or Färe-Primont
indexes; the reference observation number must be provided as a numeric value in col-
umn B of the same row as the Base command (the default value of Base is 1).
CRS to impose constant returns to scale (the default is variable returns to scale)
NoTechChange to prohibit technical change.
NoTechRegress to prohibit technical regress.
UnitMeans to rescale the data (i.e., change units of measurement) so that all output and input quan-
tity variables have unit means. This option can be used to avoid numerical problems
when quantity variables are of very different orders of magnitude. Some LP software
packages recommend that LP variables should always be measured in units such that no
value is greater than 1E+5 or less than 1E-4 (Winston, 2004, p.167). Malmquist-hs,
Malmquist-it, Hicks-Moorsteen and Färe-Primont indexes (i.e., indexes that involve
solving LPs) may be sensitive to rescaling.
Window (available in Professional edition only) to estimate the production technology using all
observations in a moving window of time periods; the window length must be provided
as a numeric value in column B of the same row as the Window command.
Figures 4 and 5 illustrate the use of some of these options. Figure 4 shows how the command rows in the input
file in Figure 3 should be modified to compute and decompose Paasche TFP indexes under the assumption of
CRS. Observe that it is possible to add text to cells in columns D and E of any command row (i.e., any row
before the end command). Figure 5 is a partial screenshot of the file NE_input.csv used for analysing a subset
of the US agricultural data compiled by Ball, Hallahan and Nehring (2004).
Running the Executable File
The DPIN executable file is DPIN.exe. Double-clicking this file will open windows similar to those depicted in
Figures 6 to 8. Figure 6 is a command window that reports program and run-time information; Figure 7 is a
window used to browse and select the input file; and Figure 8 is the command window as it appears shortly after
the input file NE_input.csv has been selected. DPIN performs six sets of computations in the decomposition of
TFP indexes, and the command window reports how the program is progressing through each of these sets.
Warnings and common runtime errors are also reported in the command window.
15
Figure 4. Example Input File for Computing and Decomposing Paasche TFP Indexes
Figure 5. Input File for Analysing Productivity in the Northeast (NE) Farm Production Region of the United States
16
Figure 6. Initial Command Window
Figure 7. Input File Selection Window
Figure 8. Command Window After Data Processing
17
7. DPIN OUTPUT FILES
DPIN computes and reports estimated levels of TFP and various types efficiency for every firm in every time
period in the sample. It also computes and reports indexes of TFP change and efficiency change. DPIN results
are written to EXCEL output files having the following extensions (and contents):
_output_indexes.csv This file reports indexes measuring changes in aggregate output (labelled dQ) ,
aggregate input (dX), total factor productivity (dTFP), the technology (dTech
= dTFP*) and various types of efficiency (dTFPE, dOTE, dOSE, dOME,
dROSE, dOSME, dITE, dIME, dRISE, dISME and dRME). If prices are
available then the Professional Edition of DPIN also reports indexes measur-
ing changes in revenue (dRev), cost (dCost), profitability (dProf), the aggre-
gate output price (dP), the aggregate input price (dW) and the terms of trade
(dTT). If an intransitive index (e.g., Hicks-Moorsteen) is used then DPIN re-
ports comparisons between firm i in period t and firm i in period t-1 (these are
the only comparisons that are meaningful – see Section 2). If a transitive in-
dex (e.g., Färe-Primont) is used then the default comparison is between firm i
in period t and firm 1 in period 1 (this reference observation can be easily
changed using the Base command).
_output_levels.csv This file reports estimated levels of aggregate output (Q), aggregate input (X),
total factor productivity (TFP), the maximum TFP possible in each period
(TFP*) and various types of efficiency (TFPE, OTE, OSE, OME, ROSE,
OSME, ITE, IME, RISE, ISME and RME). Again, if prices are available then
the Professional Edition of DPIN also reports revenue (Rev), cost (Cost), prof-
itability (Prof), the aggregate output price (P), the aggregate input price (W)
and the terms of trade (TT). Note that if an intransitive index (e.g., Hicks-
Moorsteen) is used then it is not generally possible to use the reported esti-
mates of Q, X, TFP, TFP*, OME, ROSE, OSME, IME, ISME or RME to ob-
tain the corresponding index values reported in the _output_indexes.csv file
(see the example below).
_output_shadowprices.csv (Professional edition only) This file reports the estimated shadow prices given
by equations (86) and (87) (labelled pstar1, ..., pstarJ and wstar1, ..., wstarK).
_output_xtras.csv (Professional edition only) This file reports selected computations that may be
useful for diagnostic purposes. Specifically, it reports estimates of the aggre-
gate quantities ˆ ,itQ ,itQ ,itQ ˆ ,itX ,itX ,itX ˆ ,hsQ ,hsQ ,hsQ ˆ ,hsX hsX and hsX (la-
belled QHATt, QBARt, Qt, XHATt, XBARt, Xt, QHATs, QBARs, Qs,
XHATs, XBARs and Xt respectively) as well as estimates of ( , , ),I it hsD x q s
( , , ),I hs itD x q t ( , , ),I it itD x q t ( , , ),I it itH x q t ( , , ),O hs itD x q s ( , , ),O it hsD x q t
18
( , , )O it itD x q t and ( , , )o it itH x q t (labelled DItss, DIstt, DIttt, HIttt, DOsts,
DOtst, DOttt and HOttt respectively). If Malmquist-hs, Malmquist-it or
Hicks-Moorsteen indexes are used then the file reports the values of and
that solve LPs (51) to (54). If Färe-Primont indexes are used then the file re-
ports estimates of 0 0 0( , , )OD x q t and 0 0 0( , , )ID x q t (labelled DO000 and
DI000 respectively), the values of 0 , 0 , 0 , 0 , 0 and 0 that solve
(55) and (56), and the vectors of output and input shadow prices given by
equations (88) and (89) (labelled pstar0 and wstar0).
8. EXAMPLES
To illustrate the decomposition of intransitive TFP indexes, Figures 9 to 12 present partial screenshots of the
output files obtained after running DPIN on the input file Eg1_input.csv presented in Figure 4. This input file
instructs DPIN to compute and decompose a Paasche TFP index. Caution must be exercised when interpreting
reported levels of productivity and efficiency associated with intransitive indexes like the Paasche. Consider the
Paasche TFP index 41,42TFP that compares firm 4 in period 2 with firm 4 in period 1. The computations
involved in computing this index are as follows:
42 42 42 (2)(10) 20Q p q (cell F11 in Fig. 9 and cell J10 in Fig. 10)
42 42 42 (2)(40) (6)(20) 200X w x (cell I11 in Fig. 9 and cell K10 in Fig. 10)
41 42 41 (2)(10) 20Q p q (cell L11 in Fig. 9)
41 42 41 (2)(50) (6)(20) 220X w x (cell O11 in Fig. 9)
41,42 42 41/ 20 / 20 1Q Q Q (cell J10 in Fig. 11)
41,42 42 41/ 200 / 220 0.9091X X X (cell K10 in Fig. 11)
41,42 41,42 41,42/ 1/ 0.9091 1.1TFP Q X (cell L10 in Fig. 11)
Note that the base period aggregate output 41 20Q and the base period aggregate input 41 220X are not
reported in Fig. 9 (we might expect to see them in row 6, but these entries are aggregates computed using
different aggregator functions that use period 1 prices as weights). This illustrates that reported levels of output,
input and productivity associated with (intransitive) Paasche, Laspeyres, Fisher, Malmquist-hs, Malmquist-it
and Hicks-Moorsteen indexes cannot be blindly used to construct measures of output, input and productivity
change – if an intransitive index is used then it is generally only meaningful to compare values that are
reported in the same row of the DPIN output files. For example, it is meaningful to use the entries in row 8
of Figure 10 as follows:
42 42 42 (1)(20) 20REV P Q
42 42 42 (1)(200) 200COST W X
42 42 42/ 1/1 1TT P W
42 42 42/ 20 / 200 0.1TFP Q X
19
*
42 42 2/ 0.1/ 0.1333 0.75TFPE TFP TFP
42 42 42 0.75 1 0.75TFPE OTE OSME
42 42 42 1 1 1OSME RAE ROSE
The finding that 42 0.75TFPE tells us that the productivity of firm 4 is 25% less than the maximum productiv-
ity that is possible using the technology available in period 2. The finding that 42 0.75OTE and 42 1OSME
tells us that the entire productivity shortfall is due to technical inefficiency. In terms of productivity change, it is
meaningful to use the entries in Row 8 of Figure 9 as follows:
41,42 41,42 41,42/ 2 / 0.6897 2.9PROF REV COST
41,42 41,42 41,42 2.6364 1.1 2.9PROF TT TFP
41,42 41,42 41,42/ 2 / 0.7586 2.6364TT P W
41,42 41,42 41,42/ 1/ 0.9091 1.1TFP Q X
* *
41,42 2 1 41,42 41,42( / ) 1 1.1 1.1TFP TFP TFP TFPE dTech TFPE
41,42 41,42 41,42 1 1.1 1.1TFPE OTE OSME
41,42 41,42 41,42 1 1.1 1.1OSME RAE ROSE
Thus, we find that the profitability of firm 4 has increased almost three-fold 41,42( 2.9)PROF as a result of a
significant improvement in the terms of trade 41,42( 2.6364)TT and a 10% increase in productivity
41,42( 1.1).TFP The improvement in the terms of trade is due to a doubling of output prices 41,42( 2)P and a
25% fall in input prices 41,42( 0.7586).W Finally, all of the increase in productivity is due to an increase in
residual output-oriented scale efficiency 41,42( 1.1)ROSE (a residual measure that captures productivity changes
associated with changes in both inputs and outputs).
To illustrate the decomposition of transitive TFP indexes, Figures 13 to 16 present partial screenshots of the
output files obtained after running DPIN on the input file NE_input.csv presented in Figure 5. The Färe-
Primont index is transitive, so it is meaningful to compare cells in any columns or rows of these tables. For
example, the TFP of state 6 in 1960 is 0.5891 (cell L3 in Fig. 13) and the TFP of state 19 in 1960 is 0.5499 (cell
L7 in Fig. 13). Thus, the TFP index that compares state 6 with state 19 is 0.5891/0.5499 = 1.0714 (cell L3 in
Fig. 14). Observe from row 24 in Figure 13 that state 44 achieved the maximum TFP possible in 1961 (indeed,
because there is no technical change in the first three years, state 44 achieved the maximum TFP possible in any
of the years 1960-1962). Thus, the efficiency scores reported in Table 1 can all be viewed as indexes that
compare efficiency in each state in each year with the efficiency of state 44 in 1961. When it comes to an
examination of the components of productivity change, it is convenient to use the Data Sort facility in EXCEL
to sort the results first by state and then by year, and to then use the Insert Line (graph) option to plot different
measures of interest. For example, Figures 17 and 18 present decompositions of profitability change and TFP
change in state 6 over the period 1960 to 1989. For further examples of the computation and interpretation of
transitive TFP indexes, see O'Donnell, Fallah-Fini and Triantis (2011) (US interstate highway maintenance),
O'Donnell (2010b) (US agriculture) and O'Donnell (2011b) (US manufacturing sectors).
20
9. RUNTIME ERRORS
Error diagnostics are written to text files having extensions _output_xxerrors.txt. Common error codes reported
in these files (and the command window) are:
-1 this error code indicates that a linear program cannot be solved. Some program may fail to solve
because of numerical errors that occur when input and output variables are of very different orders
of magnitude (e.g., some variables are measured in thousands of units and others are measured in
tenths of a unit). Numerical errors often lead to values of 1.#IND or 1.#INF in the DPIN output
file and can generally be avoided by scaling all input and output quantity variables to have unit
means (use the UnitMeans command). Other linear programs may fail to solve simply because
they are infeasible – see O'Donnell (2010a).
5 this error code indicates that the maximum number of simplex iterations has been exceeded. The
maximum number of iterations may be reached if the linear program is degenerate. For details on
degeneracy and cycling in linear programs see Winston (2004, p. 168-171).
Note that the Hicks-Moorsteen index sometimes takes the value zero. This is not an error – the Hicks-
Moorsteen index that compares the TFP of firm i in period t with the TFP of firm i in period s will always take
the value zero when, for example, firm i uses less of one input in period t than any other firm in the sample, and
if it fails to produce an output in period t that it had produced in period s. Finally, values of 1.#IND or 1.#INF
mean that DPIN has either exceeded the finite limits of floating point arithmetic (e.g., generated a number that is
infinitely large) or attempted to obtain a result that is simply undefined (e.g. division by zero). In such cases it
is often worthwhile checking for outlier observations (e.g., observations where all inputs are zero or where one
or more variables are extremely large).
Figure 9. Paasche Example: Selected Computations (Eg1_output_xtras.csv)
Figure 10. Paasche Example: Levels of Productivity and Efficiency (Eg1_output_levels.csv)
22
Figure 11. Paasche Example: Indexes of Productivity and Efficiency Change Under CRS (Eg1_output_indexes.csv)
Figure 12. Paasche Example: Revenue-deflated Output Shadow Prices and Cost-deflated Input Shadow Prices (Eg1_output_shadowprices.csv)
23
Figure 13. NE Farm Production Example: Levels of Productivity and Efficiency (NE_output_levels.csv)
24
Figure 14. NE Farm Production Example: Indexes of Productivity and Efficiency Change Relative to State 19 in 1960 (NE_output_indexes.csv)
Figure 15. NE Farm Production Example: Revenue-deflated Output Shadow Prices and Cost-deflated Input Shadow Prices (NE_output_shadow prices.csv)
25
Figure 16. NE Farm Production Example: Selected Computations (NE_output_xtras.csv)
Figure 17. NE Farm Production Example: The Components of Profitability Change in State 6
Figure 18. NE Farm Production Example: The Components of TFP Change in State 6
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
1960
1962
1964
1966
1968
1970
1972
1974
1976
1978
1980
1982
1984
1986
1988
dProf
dTT
dTFP
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
1960
1962
1964
1966
1968
1970
1972
1974
1976
1978
1980
1982
1984
1986
1988
dTFP
dTech
dOTE
dOSME
9. REFERENCES
Balk, B. M. (1998). Industrial Price, Quantity, and Productivity Indices: The Micro-Economic Theory and an
Application. Boston, Kluwer Academic Publishers.
Ball, V. E., C. Hallahan and R. Nehring (2004). "Convergence of Productivity: An Analysis of the Catch-Up
Hypothesis Within a Panel of States." American Journal of Agricultural Economics 86(5): 1315-1321.
Bjurek, H. (1996). "The Malmquist Total Factor Productivity Index." Scandinavian Journal of Economics 98(2):
303-313.
Caves, D. W., L. R. Christensen and W. E. Diewert (1982). "The Economic Theory of Index Numbers and the
Measurement of Input, Output, and Productivity." Econometrica 50(6): 1393-1414.
Diewert, W. E. (1992). "Fisher Ideal Output, Input, and Productivity Indexes Revisited." Journal of Productivity
Analysis 3: 211-248.
Färe, R. and D. Primont (1995). Multi-output Production and Duality: Theory and Applications. Boston, Kluwer
Academic Publishers.
Farrell, M. J. (1957). "The Measurement of Productive Efficiency." Journal of the Royal Statistical Society,
Series A (General) 120(3): 253-290.
Grosskopf, S., D. Margaritis and V. Valdmanis (1995). "Estimating output substitutability of hospital services:
A distance function approach." European Journal of Operational Research 80(1995): 575-587.
Hicks, J. R. (1961). "Measurement of Capital in Relation to the Measurement of Other Economic Aggregates"
in Lutz, F. A. and D. C. Hague (eds.) The Theory of Capital. London, Macmillan.
Lowe, J. (1823). The Present State of England in Regard to Agriculture, Trade and Finance. London, Longman,
Hurst, Rees, Orme and Brown.
Moorsteen, R. H. (1961). "On Measuring Productive Potential and Relative Efficiency." The Quarterly Journal
of Economics 75(3): 151-167.
O'Donnell, C. J. (2008). "An Aggregate Quantity-Price Framework for Measuring and Decomposing
Productivity and Profitability Change." Centre for Efficiency and Productivity Analysis Working
Papers WP07/2008. University of Queensland.
http://www.uq.edu.au/economics/cepa/docs/WP/WP072008.pdf.
O'Donnell, C. J. (2010a). "Measuring and Decomposing Agricultural Productivity and Profitability Change."
Australian Journal of Agricultural and Resource Economics 54(4): 527-560.
O'Donnell, C. J. (2010b). "Nonparametric Estimates of the Components of Productivity and Profitability Change
in U.S. Agriculture." Centre for Efficiency and Productivity Analysis Working Papers WP02/2010.
University of Queensland. http://www.uq.edu.au/economics/cepa/docs/WP/WP022010.pdf.
O'Donnell, C. J. (2011a). "Econometric Estimation of Distance Functions and Associated Measures of
Productivity and Efficiency Change." Centre for Efficiency and Productivity Analysis Working Papers
WP01/2011. University of Queensland.
http://www.uq.edu.au/economics/cepa/docs/WP/WP012011.pdf.
O'Donnell, C. J. (2011b). "The Sources of Productivity Change in the Manufacturing Sectors of the U.S.
Economy." Centre for Efficiency and Productivity Analysis Working Papers WP07/2011. University of
Queensland. http://www.uq.edu.au/economics/cepa/docs/WP/WP072011.pdf.
28
O'Donnell, C. J., S. Fallah-Fini and K. Triantis (2011). "Comparing Firm Performance Using Transitive
Productivity Index Numbers in a Meta-Frontier Framework." Centre for Efficiency and Productivity
Analysis Working Papers WP08/2011. University of Queensland.
http://www.uq.edu.au/economics/cepa/docs/WP/WP082011.pdf.
Shephard, R. W. (1953). Cost and Production Functions. Princeton, Princeton University Press.
Winston, W. L. (2004). Operations Research: Applications and Algorithms. Belmont CA, Brooks Cole.